My expertise with medieval Europe is too limited to respond to your question with respect to what you probably intended when you said "medieval". I do know my way around the medieval Islamic world, where games of chance and skill played in public are a very common trope of stories and poetry.

These might take the form of a competition among reciters of poetry; a debate on scripture or law; storytelling; the playing of traditional instruments; singing; acrobatics; animal handling; or many others. The Qur'an frowns upon gambling (it's complex) and so it's seen as more "sinful" than it would be here and now. So there was usually an element of skill in these public contests, whereas real gambling would go on behind closed doors as was proper.

The closest analogue I can think of to what you're talking about would be getting a bunch of star seminary students in a public house or square and quiz them on the religious texts: in no way the most common sort of public contest, but still familiar enough to most. The prizes would likely be cash, or maybe fancy clothes.

Though now every time I lecture students about this period, I am going to use the phrase BRAND NEW CAMEL, so thanks OP for that one.

Since I can think of an obvious answer but it would only be a guess on my part, I'll use the form of a question. Since the op said before television, were there game shows on the radio in the pre-television era? What was the first game show?